blackVariational inference is a machine learning method for approximate inference, and it has been widely used to approximate posterior densities for Bayesian models [1]}, [2]}, [3]}. In variational inference, a trial density function is chosen and optimized by minimizing the Kullback-Leibler
(KL) divergence between the trial function and the true a posteriori function. Instead of using fully factorized trial functions where all variables are assumed to be independent (thereby likely resulting in poor approximations), more structured factorizations can be used, leading to SVI algorithms. With graphical models, SVI blackcan be formulated as message-passing algorithms [2]}, [3]}, [6]}, which is termed SVMP. In this work, SVMP is adopted because the use of SVMP facilitates the incorporation of the message passing algorithm UAMP into blackSVMP. We will show how UAMP can be used to handle the most computational intensive part of message computations, blackenabling us to achieve low complexity and high robustness. With SVMP, we can find an approximation to the marginal distribution \(p(\mathbf {x}| \mathbf {r})\) , where an approximation to \(p(\mathbf {\gamma }| \mathbf {r})\)  is also involved (the approximate inference for \(\mathbf {x}\)  and \(\mathbf {\gamma }\)  is performed alternately).
